Top Rated by Parents/Students Nationwide

07478355331

Sine & Cosine Rules

Sine Rule

What is the sine rule?

  • The sine rule allows us to find missing side lengths or angles in non– right-angled triangles
  • It states that for any triangle with angles A, B and C

fraction numerator a over denominator sin space A blank end fraction equals blank fraction numerator b over denominator sin space B end fraction equals blank fraction numerator c over denominator sin space C blank end fraction

    • Where
      • a is the side opposite angle A
      • b is the side opposite angle B
      • c is the side opposite angle C
  • You not need to remember it
  • Sin 90° = 1 so if one of the angles is 90° this becomes SOH from SOHCAHTOA

Non-Right-Angled Triangles Diagram 1a

How can we use the sine rule to find missing side lengths or angles?

  • The sine rule can be used when you have any opposite pairs of sides and angles
  • Always start by labelling your triangle with the angles and sides
    • Remember the sides with the lower-case letters are opposite the angles with the equivalent upper-case letters
  • Use the formula in the formula booklet to find the length of a side
  • To find a missing angle you can rearrange the formula and use the form

fraction numerator sin space A blank over denominator a end fraction equals blank fraction numerator sin space B blank over denominator b end fraction equals blank fraction numerator sin space C blank over denominator c end fraction

  • Substitute the values you have into the formula and solve
    • This will always give you an acute angle
    • If you know the angle is obtuse then subtract this value from 180

Exam Tip

  • Remember to check that your calculator is in degrees mode!

Worked example

The following diagram shows triangle ABC.  AB space equals space 8.1 space cm, BC space equals space 12.3 space cm, straight B straight C with hat on top straight A equals 27 degree . The angle  straight B straight A with hat on top straight C is acute.

3-3-2-sine-rule-we-question

Use the sine rule to calculate the value of:

i)
x,

  3-3-2-ai-sl-sine-rule-we-solution-i

ii)
y.

3-3-2-ai-sl-sine-rule-we-solution-ii

Cosine Rule

What is the cosine rule?

  • The cosine rule allows us to find missing side lengths or angles in non– right-angled triangles
  • It states that for any triangle

a squared equals b squared plus c squared minus 2 b c space cos space A   ;      cos space A blank equals blank fraction numerator b to the power of 2 blank end exponent plus blank c squared minus blank a squared over denominator 2 b c end fraction

    • Where
      • a is the side opposite angle A
      • b and c are the other two sides
  • You are not given either formula
    • You could memorise both of them
    • Or you could memorise the first one and rearrange it each time you use it
  • Cos 90° = 0 so if A = 90° this becomes Pythagoras’ Theorem

How can we use the cosine rule to find missing side lengths or angles?

  • The cosine rule can be used when you have two sides and the angle between them or all three sides
  • Always start by labelling your triangle with the angles and sides
    • Remember the sides with the lower-case letters are opposite the angles with the equivalent upper-case letters
  • Use the formula  a squared space equals space b squared space plus space c squared space minus space 2 b c space cos A to find an unknown side
  • Use the formula cos space A blank equals blank fraction numerator b to the power of 2 blank end exponent plus blank c squared space minus blank a squared over denominator 2 b c end fraction  to find an unknown angle
    • A is the angle between sides b and c
  • Substitute the values you have into the formula and solve

Exam Tip

  • Remember to check that your calculator is in degrees mode!

Exam Tip

The following diagram shows triangle ABC . AB space equals space 4.2 space km ,  BC space equals space 3.8 space km, AC space equals space 7.1 space km.

3-3-2-cosine-rule-we-question

Calculate the value of angle ABC.

4-11-1-cosine-rule-new-we-solution

Number Toolkit
  • Mathematical Operations
  • Negative Numbers
  • Money Calculations
  • Number Operations
  • Related Calculations
  • Counting Principles
Prime Factors, HCF & LCM
  • Types of Number
  • Prime Factor Decomposition
  • HCF & LCM
Powers, Roots & Standard Form
  • Powers, Roots & Indices
  • Standard Form
Fractions
  • Basic Fractions
  • Operations with Fractions
Percentages
  • Basic Percentages
  • Working with Percentages
Simple & Compound Interest, Growth & Decay
  • Interest & Depreciation
  • Exponential Growth & Decay
Fractions, Decimals & Percentages
  • Converting between FDP
  • Converting between FDP
Rounding, Estimation & Bounds
  • Rounding & Estimation
  • Bounds
Surds
  • Simplifying Surds
  • Rationalising Denominators
Using a Calculator
  • Using a Calculator
Algebra Toolkit
  • Algebraic Notation & Vocabulary
  • Algebra Basics
Algebraic Roots & Indices
  • Algebraic Roots & Indices
Expanding Brackets
  • Expanding Single Brackets
  • Expanding Multiple Brackets
Factorising
  • Factorising
  • Factorising Quadratics
  • Quadratics Factorising Methods
Completing the Square
  • Completing the Square
Rearranging Formulae
  • Rearranging Formulae
Algebraic Proof
  • Algebraic Proof
Linear Equations
  • Solving Linear Equations
Solving Quadratic Equations
  • Solving Quadratic Equations
  • Quadratic Equation Methods
Simultaneous Equationsr
  • Simultaneous Equations
Iteration
  • Iteration
Forming & Solving Equations
  • Forming Equations
  • Equations & Problem Solving
Functions
  • Functions Toolkit
  • Composite & Inverse Functions
Coordinate Geometrys
  • Coordinates
  • Coordinate Geometry
Linear Graphs y = mx + c
  • Straight Line Graphs (y = mx + c)
  • Parallel & Perpendicular Lines
Graphs of Functions
  • Types of Graphs
  • Graphical Solutions
  • Trig Graphs
Equation of a Circle
  • Equation of a Circle
  • Equation of a Tangents
Estimating Gradients & Areas under Graphs
  • Finding Gradients of Tangents
  • Finding Areas under Graphs
Real-Life Graphs
  • Distance-Time & Speed-Time Graphs
  • Conversion Graphs
  • Rates of Change of Graphs
Solving Inequalities
  • Solving Linear Inequalities
  • Conversion Graphs
  • Solving Quadratic Inequalities
Graphing Inequalities
  • Graphing Inequalities
Transformations of Graphs
  • Reflections of Graphs
Sequences
  • Introduction to Sequences
  • Types of Sequences
  • Linear Sequences
  • Quadratic Sequences
Ratio Toolkit
  • Simple Ratio
  • Working with Proportion
Ratio Problem Solving
  • Ratios & FDP
  • Multiple Ratios
Direct & Inverse Proportions
  • Direct & Inverse Proportion
Standard & Compound Units
  • Time
  • Unit Conversions
  • Compound Measures
Exchange Rates & Best Buys
  • Exchange Rates & Best Buys
Geometry Toolkit
  • Symmetry
  • 2D & 3D Shapes
  • Plans & Elevations
Angles in Polygons & Parallel Lines
  • Basic Angle Properties
  • Angles in Polygons
  • Angles in Parallel Lines
Bearings, Scale Drawing, Constructions & Loci
  • Bearings
  • Scale & Maps
  • Constructing Triangles
  • Constructions & Loci
Circle Theorems
  • Angles at Centre & Semicircles
  • Chords & Tangents
  • Cyclic Quadrilaterals
  • Segment Theorems
  • Circle Theorem Proofs
Area & Perimeter
  • Area & Perimeter
  • Problem Solving with Areas
Circles, Arcs & Sectors
  • Area & Circumference of Circles
  • Arcs & Sectors
Volume & Surface Area
  • Volume
  • Surface Area
Congruence, Similarity & Geometrical Proof
  • Congruence
  • Similarity
  • Geometrical Proof
Area & Volume of Similar Shapes
  • Similar Area & Volumes
Right-Angled Triangles – Pythagoras & Trigonometry
  • Pythagoras Theorem
  • Right-Angled Trigonometry
  • Exact Trig Values
Sine, Cosine Rule & Area of Triangles
3D Pythagoras & Trigonometry
  • 3D Pythagoras & Trigonometry
Vectors
  • Introduction to Vectors
  • Working with Vectors
Transformations
  • Translations
  • Reflections
  • Rotations
  • Enlargements
  • Combination of Transformations
Probability Toolkit
  • Basic Probability
  • Relative & Expected Frequency
Simple Probability Diagrams
  • Two Way Tables
  • Frequency Trees
  • Set Notation & Venn Diagrams
Tree Diagrams
  • Tree Diagrams
Combined & Conditional Probability
  • Combined Probability
  • Conditional Probability
  • Combined Conditional Probabilities
Statistics Toolkit
  • Mean, Median & Mode
  • Averages from Tables
  • Range & Quartiles
  • Comparing Distributions
  • Population & Sampling
Statistical Diagrams
  • Bar Charts & Pictograms
  • Pie Charts
  • Time Series Graphs
  • Working with Statistical Diagrams
Histograms
  • Histograms
Cumulative Frequency & Box Plots
  • Cumulative Frequency Diagrams
  • Box Plots
Scatter Graphs & Correlation
  • Scatter Graphs

Leading tutoring agency within the North East, that specialises in getting students top grades all over the country.

Quick Links

Social

Get In Touch

Get Started

Book A Demo

  • 2024 League Of Learning